Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$?

2012-09-17T18:46:47Z

Does there exist a matroid that is representable over $\mathbb{R}$ but not over $\mathbb{Q}$?

In particular, can one give a positive answer using a nonrational polytope, i.e., a combinatorial polytope that cannot be realized as the convex hull of rational vertices? (Such things do exist; see, e.g., p.94 of Grünbaum's Convex Polytopes.) The vertex sets of the faces of a convex polytope certainly form the flats of a matroid, but it's not clear to me why the same matroid could not be realized by affine dependences of a set of points not in convex position.

Answer by Igor Pak for Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$?

2012-09-17T19:07:33Z

Jeremy, on the very same page 94 you will find a "point and line configuration" called Perles configuration which when viewed as set ov vectors in $\Bbb R^3$ is a matroid that is realizable over $\Bbb Q[\sqrt{5}]$ but not over $\Bbb Q$. In my book I even prove it (Ex 12.3) - sorry to make a plug, this is the only place with a proof I know.

Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$? - MathOverflow

Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$?

Answer by Igor Pak for Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$?